Iterative Auction Design for Graphical Valuations Part I: Tree Valuations∗

In this work, we design simple and e cient iterative auctions for selling multiple items in settings where bidders’ valuations belong to a special class (tree valuations) that can exhibit both value complementarity and substitutability. Our first contribution is to provide a compact linear programming formulation of the e cient allocation problem and use it to establish the existence of a Walrasian equilibrium for tree valuations that satisfy an additional technical condition. This result reveals a new class of valuations for which a Walrasian equilibrium exists in the presence of value complementarities. We then provide an iterative algorithm that can be used for the solution of this linear programming formulation. Complementing the algorithm with an appropriate payment rule, we obtain an iterative auction which implements the e cient outcome (at an ex-post perfect equilibrium). This auction relies on a simple pricing rule, compact demand reports, and uses a novel (interleaved) price update structure to assign final payments to bidders that guarantee truthful bidding.